Finite Group Databasebeta

← #29738899

Subgroups — #29738899

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456349C211yesmaximal, characteristic, complemented
2#21455674C211yesmaximal, characteristic
3#21455670C211yesmaximal, characteristic
4#21454412C211yesmaximal, characteristic
5#21447328C211yesmaximal, characteristic, complemented
6#21447327C211yesmaximal, characteristic, complemented
7#21447169C211yesmaximal, characteristic, complemented
8#21447168C211yesmaximal, characteristic, complemented
9#21446598C211yesmaximal, characteristic, complemented
10#21446566C211yesmaximal, characteristic, complemented
11#21446054C211yesmaximal, characteristic
12#21445981C211yesmaximal, characteristic, complemented
13#21445876C211yesmaximal, characteristic
14#21445657C211yesmaximal, characteristic
15C61×#1860C211yesmaximal, characteristic
16C2×#14519784C2211yescharacteristic
17#16840393C2211yescharacteristic
18#16840390C2211yescharacteristic
19#16840245C2211yescharacteristic
20#16840170—21no—
21#16840169—21no—
22#16840163—21no—
23#16840159—21no—
24#16840069—21no—
25#16840067—21no—
26#16840064—21no—
27#16840062—21no—
28#16840057—21no—
29#16840055—21no—
30#16840042—21no—
31#16840040—21no—
32C2×#14519712C2211yescharacteristic
33#16839805—21no—
34#16839805C2211yescharacteristic
35#16839804C2211yescharacteristic
36#16839804C2211yescharacteristic
37#16839802—21no—
38#16839802C2211yescharacteristic
39#16839802C2211yescharacteristic
40#16839798—21no—
41#16839798C2211yescharacteristic
42C2×#14519710C2211yescharacteristic
43C2×#14519710C2211yescharacteristic
44C2×#14519708—21no—
45C2×#14519708C2211yescharacteristic
46C2×#14519708C2211yescharacteristic
47#16839790C2211yescharacteristic
48#16839789C2211yescharacteristic
49#16839763C2211yescharacteristic
50#16839758C2211yescharacteristic
51#16839720C2211yescharacteristic
52C2×C4×#92418C2211yescharacteristic
53#16839628C2211yescharacteristic
54#16839531C2211yescharacteristic
55#16839530C2211yescharacteristic
56#16839530C2211yescharacteristic
57#16839526C2211yescharacteristic
58#16839522C2211yescharacteristic
59#16839522C2211yescharacteristic
60C61×#520C2211yescharacteristic
61C61×#428C2211yescharacteristic
62C2×#104×C61C2211yescharacteristic
63C61×#417C2211yescharacteristic
64C61×#382C2211yescharacteristic
65C61×#343C2211yescharacteristic
66C61×#339C2211yescharacteristic
67#14519784—22no—
68#14519784Q814yescomplemented
69C2×#11758194—21no—
70C2×#11758194C2311yescharacteristic
71C2×#11758194C2311yescharacteristic
72C22×#92418—21no—
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75#14519777—21no—
76#14519776—21no—
77#14519776—21no—
78#14519776—21no—
79#14519769—21no—
80#14519768—21no—
81#14519768—21no—
82#14519768—21no—
83#14519756—41no—
84#14519754—41no—
85Q8×Dic61—21no—
86D4×Dic61—21no—
87D4×Dic61—41no—
88#14519736—21no—
89#14519736C2311yescharacteristic
90#14519736C2311yescharacteristic
91C2×C4⋊Dic61—21no—
92C2×C4⋊Dic61C2311yescharacteristic
93C2×C4⋊Dic61C2311yescharacteristic
94#14519712—22no—
95#14519711—21no—
96#14519711—21no—
97#14519710—21no—
98#14519710—21no—
99#14519710—21no—
100#14519710—21no—
101#14519710—21no—
102#14519710—21no—
103#14519710—21no—
104#14519710D411yescharacteristic
105#14519710D411yescharacteristic
106#14519709—21no—
107#14519709—21no—
108#14519708—21no—
109#14519708—21no—
110#14519708—21no—
111#14519708—21no—
112#14519708—21no—
113#14519708—21no—
114#14519708—21no—
115#14519708D411yescharacteristic
116#14519708D411yescharacteristic
117C4⋊#92418—21no—
118C4⋊#92418—21no—
119#14519706—21no—
120#14519706—21no—
121#14519701—21no—
122#14519701D411yescharacteristic
123#14519701D411yescharacteristic
124#14519698—21no—
125#14519698D411yescharacteristic
126#14519698D411yescharacteristic
127#14519687—21no—
128C4×#92418—22no—
129C8×Dic61—21no—
130#14519684—41no—
131C2×#41×C61C2311yescharacteristic
132C2×M16×C61C2311yescharacteristic
133C2×M16×C61C2311yescharacteristic
134C4×Q8×C61—21no—
135C4×D4×C61—21no—
136C4×D4×C61—41no—
137#117×C61—21no—
138#117×C61C2311yescharacteristic
139C2×C4⋊C4×C61—21no—
140C2×C4⋊C4×C61C2311yescharacteristic
141C2×C4⋊C4×C61C2311yescharacteristic
142C2×C42×C61C2311yescharacteristic
143#107×C61—21no—
144#106×C61—21no—
145#104×C61Q814yescomplemented
146#103×C61—21no—
147#102×C61—21no—
148#95×C61—41no—
149#41×C61—21no—
150#41×C61—21no—
151#41×C61C2×Q812yes—
152#41×C61C2×Q812yes—
153C2×Q8×C61C2×D411yescharacteristic
154C2×D4×C61—21no—
155C2×D4×C61C2×D411yescharacteristic
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—41no—
161C22×C4×C61C2411yescharacteristic
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—41no—
165C2×Dic122—41no—
166C2×Dic122—41no—
167M16×C61—21no—
168M16×C61—21no—
169M16×C61—21no—
170M16×C61C2×Q812yes—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—41no—
178C4⋊C4×C61C2×D411yescharacteristic
179C4⋊C4×C61C2×D411yescharacteristic
180C22⋊C4×C61—21no—
181C22⋊C4×C61—41no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—41no—
185C42×C61C2×Q812yes—
186C22⋊Dic61—21no—
187C22⋊Dic61—21no—
188C22⋊Dic61—41no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—41no—
194C4⋊Dic61—41no—
195C4⋊Dic61—41no—
196C4⋊Dic61C2×D411yescharacteristic
197C4⋊Dic61C2×D411yescharacteristic
198C4⋊Dic61C2×D411yescharacteristic
199C4⋊Dic61C2×D411yescharacteristic
200#11758196—41no—
201#11758196—41no—
202#11758196—41no—
203#11758196—41no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—41no—
207#11758194—21no—
208#11758194—24no—
209#11758194—21no—
210#11758194—21no—
211#11758194C2×Q812yes—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—24no—
216C2×#92418—21no—
217C2×#92418C2×D411yescharacteristic
218C2×#92418C2×D411yescharacteristic
219C2×#92418C2×D411yescharacteristic
220C2×#92418C2×D411yescharacteristic
221C2×#92418C2×Q812yes—
222C2×#92418#4112yes—
223C23×C61—21no—
224C23×C61H(5,𝔽2)11yescharacteristic
225Q8×C61—21no—
226Q8×C61C4⋊Q812yes—
227D4×C61—22no—
228D4×C61—22no—
229D4×C61—21no—
230D4×C61C4⋊Q812yes—
231C2×C4×C61—21no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—22no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—81no—
242C2×C4×C61—81no—
243C2×C4×C61—161no—
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×Q811yescharacteristic
247C2×C4×C61C22×Q811yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C612−1+411yescharacteristic
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—81no—
257C2×Dic61—81no—
258C2×Dic61—161no—
259Dic122—161no—
260Dic122—161no—
261C8×C61—22no—
262C8×C61—41no—
263#92418—22no—
264#92418—21no—
265#92418—21no—
266#92418—41no—
267#92418C22⋊Q812yes—
268#92418C22⋊Q812yes—
269#26080—611nomaximal, Sylow
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—22no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61#54411yescharacteristic
276C22×C61D4211yescharacteristic
277C22×C61#55411yescharacteristic
278C4×C61—22no—
279C4×C61—42no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—81no—
283C4×C61—81no—
284C4×C61C2×C22⋊Q811yescharacteristic
285C4×C61C2×C4⋊Q811yescharacteristic
286C4×C61#54711yescharacteristic
287C4×C61D4×Q811yescharacteristic
288Dic61—41no—
289Dic61—41no—
290Dic61—41no—
291Dic61—41no—
292Dic61—81no—
293Dic61—81no—
294#3163—611no—
295#3083—611no—
296#3083—611no—
297#2912—611no—
298#2089—611no—
299#2089—611no—
300#2061—611no—
301#2061—611no—
302#1951—611no—
303#1945—611no—
304#1860—611no—
305#1860D6111yescharacteristic
306#1846—611no—
307#1815—611no—
308#1764—611no—
309C2×C61—21no—
310C2×C61—21no—
311C2×C61—42no—
312C2×C61#244011yescharacteristic
313C2×C61#329011yescharacteristic
314C2×C61#329011yescharacteristic
315C2×#131—611no—
316#544—611no—
317#544—611no—
318#520—611no—
319#520C2×D6111yescharacteristic
320#501—1221no—
321#501—1221no—
322#499—1221no—
323#498—1221no—
324#479—1221no—
325#479—1221no—
326#478—1221no—
327#478—1221no—
328#477—1221no—
329#477—1221no—
330#474—1221no—
331#474—1221no—
332C2×#108—611no—
333#428—611no—
334#428—611no—
335#428—611no—
336#428—611no—
337#428—611no—
338#428—1221no—
339#428—1221no—
340#428C2×D6111yescharacteristic
341#427—611no—
342#427—1221no—
343C2×#107—611no—
344C2×#107—611no—
345C2×#106—611no—
346C2×#106—611no—
347C2×#106—1221no—
348#424—611no—
349#424—611no—
350C2×#104C2×D6111yescharacteristic
351#417—611no—
352#417C2×D6111yescharacteristic
353#416—611no—
354#405—611no—
355C2×C4×C8—611no—
356#382—611no—
357#382C2×D6111yescharacteristic
358#343—611no—
359#343—611no—
360#343—611no—
361#343C2×D6111yescharacteristic
362#341—611no—
363#339—611no—
364#339—611no—
365#339C2×D6111yescharacteristic
366C61#2608011yescharacteristic, complemented, Sylow
367C2×#41C22×D6111yescharacteristic
368#131—614no—
369#131—1222no—
370C2×M16—611no—
371C2×M16—611no—
372C2×M16—1221no—
373C2×M16C22×D6111yescharacteristic
374C2×M16C22×D6111yescharacteristic
375C22×C8—611no—
376C22×C8—611no—
377C22×C8—1221no—
378C4⋊Q8—1221no—
379C4⋊Q8—1221no—
380C4⋊Q8—1221no—
381C4⋊Q8—1221no—
382#125—1221no—
383#125—1221no—
384#125—1221no—
385#125—1221no—
386C22⋊Q8—2441no—
387C22⋊Q8—2441no—
388C4×Q8—21no—
389C4×Q8—1221no—
390C4×D4—21no—
391C4×D4—41no—
392C4×D4—1221no—
393C4×D4—2441no—
394#117—21no—
395#117—611no—
396#117—611no—
397#117—1221no—
398#117C22×D6111yescharacteristic
399C2×C4⋊C4—21no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C4⋊C4C22×D6111yescharacteristic
405C2×C42C22×D6111yescharacteristic
406#108—1222no—
407#107—21no—
408#107—611no—
409#107—611no—
410#107—1221no—
411#107—1221no—
412#107—1221no—
413#107—1221no—
414#107—1221no—
415#107—1221no—
416#107—1221no—
417#107—1221no—
418#107—1221no—
419#106—21no—
420#106—611no—
421#106—611no—
422#106—1221no—
423#106—1221no—
424#106—1221no—
425#106—1221no—
426#106—1221no—
427#106—1221no—
428#106—1221no—
429#106—1221no—
430#106—1221no—
431C4⋊C8—1221no—
432C4⋊C8—1221no—
433C4⋊C8—1221no—
434C4⋊C8—1221no—
435#104Dic12214yescomplemented
436#103—21no—
437#103—611no—
438#103—611no—
439#103—1221no—
440#102—21no—
441#102—611no—
442#102—611no—
443#102—1221no—
444#97—1221no—
445C4×C8—1222no—
446C4×C8—1221no—
447#95—41no—
448#95—2441no—
449#41—21no—
450#41—21no—
451#41C2×Dic12212yes—
452#41Q8×D6112yes—
453C2×Q8—2441no—
454C2×Q8—2441no—
455C2×Q8D4×D6111yescharacteristic
456C2×D4—21no—
457C2×D4D4×D6111yescharacteristic
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—41no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—2441no—
466C22×C4C23×D6111yescharacteristic
467M16—21no—
468M16—21no—
469M16—21no—
470M16—612no—
471M16—1224no—
472M16—1221no—
473M16—1221no—
474M16—1221no—
475M16C2×Dic12212yes—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—611no—
479C2×C8—611no—
480C2×C8—612no—
481C2×C8—612no—
482C2×C8—611no—
483C2×C8—611no—
484C2×C8—1224no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—41no—
494C4⋊C4—611no—
495C4⋊C4—611no—
496C4⋊C4—611no—
497C4⋊C4—611no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—2441no—
503C4⋊C4—2441no—
504C4⋊C4—2441no—
505C4⋊C4—2441no—
506C4⋊C4—2441no—
507C4⋊C4—2441no—
508C4⋊C4—2441no—
509C4⋊C4D4×D6111yescharacteristic
510C4⋊C4D4×D6111yescharacteristic
511C22⋊C4—21no—
512C22⋊C4—41no—
513C22⋊C4—1221no—
514C22⋊C4—1221no—
515C22⋊C4—2441no—
516C42—21no—
517C42—21no—
518C42—41no—
519C42—1221no—
520C42—1221no—
521C42—2441no—
522C42C2×Dic12212yes—
523C23—21no—
524C23#1451982211yescharacteristic
525Q8—21no—
526Q8—9761no—
527Q8—9761no—
528Q8#1451977612yes—
529D4—22no—
530D4—22no—
531D4—21no—
532D4#1451977612yes—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—161no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—9761no—
551C2×C4—4881no—
552C2×C4—4881no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4C2×D4×D6111yescharacteristic
556C2×C4C2×D4×D6111yescharacteristic
557C2×C4C2×Q8×D6111yescharacteristic
558C2×C4C22×Dic12211yescharacteristic
559C2×C4#41×D6111yescharacteristic
560C2×C4#1451982411yescharacteristic
561C8—22no—
562C8—41no—
563C8—612no—
564C8—612no—
565C8—1222no—
566C8—1221no—
567C8—1221no—
568C8—2441no—
569C22—21no—
570C22—21no—
571C22—22no—
572C22—41no—
573C22—41no—
574C22#1684039111yescharacteristic
575C22#1684039411yescharacteristic
576C22#1684047111yescharacteristic
577C4—22no—
578C4—42no—
579C4—41no—
580C4—41no—
581C4—81no—
582C4—81no—
583C4—4881no—
584C4—4881no—
585C4—2441no—
586C4—2441no—
587C4—2441no—
588C4—2441no—
589C4C22⋊Q8×D6111yescharacteristic
590C4C2×#1451977611yescharacteristic
591C4#1684041011yescharacteristic
592C4D4×Dic12211yescharacteristic
593C2—21no—
594C2—21no—
595C2—42no—
596C2#2144952011yescharacteristic
597C2#2145759811yescharacteristic
598C2#2145760311yescharacteristic
subgroupG/H
centerC22#16840394
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1860C2
socleC22×C61D42
prime radicalC2×#41×C61C23
hypercenter#1860D61
hypocenterC61#26080
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1860

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21456349C2semidirect
5. #21447328C2semidirect
6. #21447327C2semidirect
7. #21447169C2semidirect
8. #21447168C2semidirect
9. #21446598C2semidirect
10. #21446566C2semidirect
12. #21445981C2semidirect
68. #14519784Q8semidirect
145. #104×C61Q8semidirect
366. C61#26080semidirect
435. #104Dic122semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214563492
#214556742
#214556702
#214544122
#214473282
#214473272
#214471692
#214471682
#214465982
#214465662
#214460542
#214459812
#214458762
#214456572
C61×#18602
#2608061